Problem: Given $ m \angle CBD = 8x - 97$, $ m \angle ABC = 9x - 116$, and $ m \angle ABD = 59$, find $m\angle ABC$. $B$ $A$ $D$ $C$
Explanation: From the diagram, we see that together ${\angle ABC}$ and ${\angle CBD}$ form ${\angle ABD}$ , so $ {m\angle ABC} + {m\angle CBD} = {m\angle ABD}$ Substitute in the expressions that were given for each measure: $ {9x - 116} + {8x - 97} = {59}$ Combine like terms: $ 17x - 213 = 59$ Add $213$ to both sides: $ 17x = 272$ Divide both sides by $17$ to find $x$ $ x = 16$ Substitute $16$ for $x$ in the expression that was given for $m\angle ABC$ $ m\angle ABC = 9({16}) - 116$ Simplify: $ {m\angle ABC = 144 - 116}$ So ${m\angle ABC = 28}$.